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Chemotaxis-haptotaxis model of cancer invasion with tissue remodeling is one of the important PDE’s systems in medicine, mathematics and biomathematics. In this paper we find the solution of chemotaxis-haptotaxis model of cancer invasion using the new homotopy perturbation method (NHPM). Then by comparing some estimated numerical result with simulation laboratory result, it shows that NHPM is an efficient and exact way for solving cancer PDE’s system.

Cancer invasion model is a so complex process which includes many biological procedures. Actually, different models in Mathematics were improved for many kind of cancer disease and also so many scientists tried to give more proper and applicable models (see [

Cancer is connected with degradation of the extracellular matrix (ECM) that is degraded with the matrix degradation enzymes (MDEs) which are concealed by colony of tumor cells. The degradation produces spatial gradients which aim movement of invasive cells through a procedure named chemotaxis (cellular locomotion aimed to response to a concentration gradient of the diffusible MDE) or through a procedure which is called haptotaxis (cellular locomotion aimed to response to an association gradient of the non-diffusible adhesive molecules within extracellular matrix). Chaplain et al. [2,3] introduced a system of partial differential equations of cancer invasion with tissue. As long as this introduction, other researchers proved uniqueness and existence of solutions associated to this PDE model [4-8].

Chemotaxis-haptotaxis model of cancer invasion with tissue remodeling [

The homotopy perturbation method (HPM) was proposed by Ji-Huan He [

This paper is arranged as follows. Section 2 describes the chemotaxis-haptotaxis model of cancer invasion of tissue. In Section 3, the new modification of HPM which called NHPM is presented. In Section 4 model is solved by NHPM. In Section 5 computed solution by NHPM is compared with simulations results.

In this section, we recall the setting of [3,18,19]. Cancer invasion model consists of the 3 variables; the matrix degrading enzyme concentration (MDE concentration)

We assume following boundary conditions presented by the equations from [

and we consider following the initial conditions

The equations describing the dynamics of each variable as follows (according to [

where

Diffusion-Production-Decay: Now we suppose dispersal cell density is altered. Then let

Proteolysis: Spread of a cancer disease relay on occupies and metastasizes (for detailed information see [20- 22]).

Chemotaxis: Chemotaxis is movement of microorganisms and cells in response and to a chemical sign and in the enclosing location in tissue [

Chemoattractants are chemicals that prefer to increase the desire to move towards a given chemical origin, while chemorepellants encourage organisms and microorganisms or cells to move in the converse and opposite direction.

Now we give an example to obtain application of chemotaxis process. This process could be both beneficial (in life phenomenon) and destructive (cause of diseases) for body. During the fertilization, chemotaxis is the main step that sexual reproduction depends on, to permit migration of sperms toward an egg, following chemoattractants which produced through the egg. Therefore chemotaxis can complete fertilization.

Chemotaxis could be interrupted by the chemical signals. Confusion and disorientation of cells is the reasons to make mistakes. Another factor to interrupt this movement is limitation by environmental factors which can lead to the errors in navigation; moving organisms toward the toxins and away from sources of nutrition. In a nerve injury moving the new growing cells (during the replacement procedure with the damaged cells) toward a wrong direction may be lead to create a cancer tissue. Also there is a considerable motivation for the researchers in all over the world to learn more about the chemotaxis and correspondence procedures, as this process is very important both in treating injuries and harsh diseases as well as addressing barrenness [see 23].

Haptotaxis: Cell haptotaxis describes cell migration toward or along a gradient of chemoattractants or adhesion sites in the extracellular matrix [24,25].

Proliferation: This rate depends on the kind of cancer disease, age and other biological parameters in patients (see complete details about proliferation in [26,27]).

The general form of a system of PDEs can be considered as the following:

Coresspondence with the following initial conditions:

where U_{j} is the solution of the jth equation, _{j} and their partial derivatives, and

For solving system (1.4) we follow the steps which are quoted from [

or

where

By applying the inverse operator,

such that

Let us present the solution of the system (1.6) as the following:

where

where

Substituting (1.7) and (1.8) into (1.6) and equating the coefficients of p with the same powers leads to

Now if we solve these equations in such a way that

Therefore the exact solution may be obtained as the following:

It is worth mentioning that if

which can be used in (1.9),

In this section we are going to solve this complicated PDEs model numerically and by using the so-called modification of the Homotopy Perturbation Method that named New Homotopy perturbation Method. We just calculate two order of the He’s polynomials but you will see how is the exactness of this way and be motivate and interested to use this numerical way in your models and systems.

By definitions of gradient

For solving this system by using NHPM, we consider the following homotopy (under this assumption which

Applying the inverse operator,

Substituting Equation (1.7) into the above equations, collecting the terms with the same powers of

According to the previous section, assume

By substituting (1.12) into (1.11),

Finally it is done and we reached to the finall solutions. Solutions preferably are written in their completed form and without any simplification for using and examinig of readers. Note that you can replace your constant parameters and initial conditions without any restrictions, because you can use you can use initial conditions, v, u, w in their function forms depending on x and their derivations to x.

Now

and therefore solutions can be expressed as follows:

In this section

All of our requiremetns for reaching to this purpose is initial conditions and positive constants. Now consider the

additionally, consider every t equal to 3 hours and numerical result of positive parameters according to [

At this stage, by using the Maple software (version 15) and plotting

As you saw in past section, we calculated and estimated our solutions just by by two steps and orders (with very long term and complicated calculations) in the NHPM, but the figures show it is sufficient for our estimation. While by the Homotopy Perturbation Method (HPM) and Adomian Decomposition Method (ADM) we had to collapse finall series to a well known or calculate another orthers and steps for a exact solution, the New Homotopy Perturbation Method (NHPM) reached to the exact solution just by two steps of our calculating.

In

u(x, t) v(x, t) w(x, t)

(a) (b) (c)

u(x, t) v(x, t)

(a) (b)

simulation’s results at

New homotopy perturbation method (NHPM) is applied to the numerical solution for solving chemotaxis-haptotaxis model of cancer invasion of tissue (a complicated nonlinear PDEs system). As it was seen, differences between simulation results and NHPM results are very small. Thus the present method is very effective and convenient. Our suggestion is to use this numerical way to solve other PDEs system for example in biology.